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This rational curve sends the zeros of G to each of the coordinate points of ; that is, all but one of the vanish for a zero of G. Conversely, any rational normal curve passing through the n + 1 coordinate points may be written parametrically in this way.

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## Some Related Sentences

rational and curve

( which did not exist

**in**Diophantus's time ), his method would**be**visualised as drawing**a**tangent**to****a**__curve__at**a**known__rational__point**,**and then finding**the**other point**of**intersection**of****the**tangent with**the**__curve__**;****that**other point**is****a**new__rational__point**.**
Analytically

**,**x can also**be**raised**to**an irrational power (**for**positive values**of**x );**the**analytic properties are analogous**to**when x**is**raised**to**__rational__powers**,****but****the**resulting__curve__**is**no longer algebraic**,**and cannot**be**analyzed via algebraic geometry**.**
These solutions yield good

__rational__approximations**of****the**form x / y**to****the**square root**of****n****.**In Cartesian coordinates**,****the**equation has**the**form**of****a**hyperbola**;**it can**be**seen**that**solutions occur where**the**__curve__has integral ( x**,**y ) coordinates**.**
It

**is**clear**that****a**0 % tax rate raises no revenue**,****but****the**Laffer__curve__hypothesis**is****that****a**100 % tax rate will also generate no revenue because at such**a**rate there**is**no longer**any**incentive**for****a**__rational__taxpayer**to**earn**any**income**,**thus**the**revenue raised will**be**100 %**of**nothing**.**
The definition

**of**elliptic__curve__from algebraic geometry**is**connected non-singular projective__curve__**of**genus**1**with**a**given__rational__point on it**.**
In 1976

**,**M**.**Rosen showed how**to**realize**any**countable abelian group as**the**class group**of****a**Dedekind domain which**is****a**subring**of****the**__rational__function field**of**an elliptic__curve__**,**and conjectured**that**such an " elliptic " construction should**be**possible**for****a**general abelian group ( Rosen 1976 ).
In number theory

**,****the**Mordell conjecture**is****the**conjecture made by**that****a**__curve__**of**genus greater than**1**over**the**field Q**of**__rational__numbers has only finitely many__rational__**points****.**
* Case g =

**1**: no**points****,**or C**is**an elliptic__curve__and its__rational__**points**form**a**finitely generated abelian group ( Mordell's Theorem**,**later generalized**to****the**Mordell – Weil theorem ).
* The Mordell conjecture

**that****a**__curve__**of**genus greater than**1**over**a**number field has only finitely many__rational__**points****;**
The theorem states

**that****any**elliptic__curve__over Q can**be**obtained via**a**__rational__map with integer coefficients from**the**classical modular__curve__
A well-known example

**is****the**Taniyama – Shimura conjecture**,**now**the**modularity theorem**,**which proposed**that****each**elliptic__curve__over**the**__rational__numbers can**be**translated into**a**modular form (**in**such**a****way**as**to**preserve**the**associated L-function ).
Rational Bézier

__curve__– polynomial__curve__defined**in**homogeneous coordinates ( blue ) and its projection on plane –__rational____curve__( red )
It applies

**to**an elliptic__curve__E**,**and**the**problem it attempts**to**solve**is****the**prediction**of****the**rank**of****the**elliptic__curve__over**the**__rational__numbers ( or another global field ): i**.**e**.****the**number**of**free generators**of**its group**of**__rational__**points****.**

rational and sends

The evaluation map

__sends__**the**fundamental class**of**M**to****a**d-dimensional__rational__homology class**in**Y**,**denoted

rational and zeros

If

**the**transfer function**is****a**__rational__function with real poles and__zeros__**,**then**the**Bode plot can**be**approximated with straight lines**.**
The existence

**of**irreducible polynomials**of**degree greater than**one**( without__zeros__**in****the**original field ) historically motivated**the**extension**of****that**original number field so**that**even these polynomials can**be**reduced into linear factors: from__rational__numbers ( ),**to****the**real subset**of****the**algebraic numbers ( ), and finally**to****the**algebraic subset**of****the**complex numbers ( ).

rational and G

Additionally

**,**consider**for**instance**the**unit circle S**,**and**the**action on S by**a**group__G__consisting**of****all**__rational__rotations**.**
*

__G__**.**W**.**F**.**Hegel: Emphasized**the**" cunning "**of**history**,**arguing**that**it followed**a**__rational__trajectory**,**even while embodying seemingly irrational forces**;**influenced Marx**,**Kierkegaard**,**Nietzsche**,**and Oakeshott**.**
Some philosophers believe

**that****the**" no-free-lunch**in**search and optimization theorem "**of**David Wolpert and William__G__**.**Macready**is****a**probability-based extension**of**induction**,**yet**this****is**misleading**,**as inductive logic accustomed**to**probabilistic arguments and**the**No free lunch theorem ( NFL )**is**more**a**variation**of**economic__rational__choice theory**.**
* The irrational numbers are

**a**__G__< sub > δ </ sub > set**in**R**,****the**real numbers**,**as they can**be****written**as**the**intersection over**all**__rational__numbers q**of****the**complement**of****;**Inverse problem

**of**Galois theory: Given

**a**group

__G__

**,**find an extension

**of**

**the**

__rational__number or other field with

__G__as Galois group

**.**

Alternatively

**,****the**quantum group U < sub > q </ sub >(__G__) can**be**regarded as an algebra over**the**field C ( q ),**the**field**of****all**__rational__functions**of**an indeterminate q over C**.**
Similarly

**,****the**quantum group U < sub > q </ sub >(__G__) can**be**regarded as an algebra over**the**field Q ( q ),**the**field**of****all**__rational__functions**of**an indeterminate q over Q ( see below**in****the**section on quantum groups at q
Another example: P being as above

**,****a**resolvent R**for****a**group__G__**is****a**polynomial whose coefficients are polynomials**in****the**coefficients**of**p**,**which provides some information on**the**Galois group**of**P**.**More precisely**,**if R**is**separable and has**a**__rational__root then**the**Galois group**of**P**is**contained**in**__G__**.**For example**,**if D**is****the**discriminant**of**P then**is****a**resolvent**for****the**alternating group**.**
That implies

**that****any**two__rational__functions F and__G__**,****in****the**function field**of****the**modular**curve****,**will satisfy**a**modular equation P ( F**,**__G__) = 0 with P**a**non-zero polynomial**of**two variables over**the**complex numbers**.**
* For

**any**non-zero invariant vector**in****a**__rational__representation on__G__**,**there**is**an invariant homogeneous polynomial**that**does not**vanish**on it**.**
Hilbert had shown

**that****this**question**is**related**to****a**rationality question**for**__G__: if K**is****any**extension**of**Q**,**on which__G__acts as an automorphism group and**the**invariant field K < sup >__G__</ sup >**is**__rational__over Q**,**then__G__**is**realizable over Q**.**
Some

**of****this****is**based on constructing__G__geometrically as**a**Galois covering**of****the**projective line:**in**algebraic terms**,**starting with an extension**of****the**field Q ( t )**of**__rational__functions**in**an indeterminate t**.**After**that****,****one**applies Hilbert's irreducibility theorem**to**specialise t**,****in**such**a****way**as**to**preserve**the**Galois group**.**
Hegelianism

**is****a**collective term**for**schools**of**thought following or referring**to**__G__**.**W**.**F**.**Hegel's philosophy which can**be**summed up by**the**dictum**that**"**the**__rational__alone**is**real ", which means**that****all**reality**is**capable**of**being expressed**in**__rational__categories**.**
Berlin contended

**that**under**the**influence**of**Plato**,**Aristotle**,**Jean-Jacques Rousseau**,**Immanuel Kant and__G__**.**W**.**F**.**Hegel**,**modern political thinkers often conflated positive liberty with__rational__action**,**based upon**a**__rational__knowledge**to**which**,**it**is**argued**,**only**a**certain elite or social group has access**.**
Applying

**a**result**of**MacIntyre on**the**model theory**of**p-adic integers**,****one**deduces again**that**ζ < sub >__G__</ sub >( s )**is****a**__rational__function**in**p < sup >− s </ sup >.
Harold Pinter writes

**that**Raymond__G__**.**H**.**Seitz: " had**a**very good reputation as**a**__rational__**,**responsible and highly sophisticated man**.**

rational and each

Since C

**is**__rational__**,****this**correspondence has K coincidences**,**__each__**of**which implies**a**line**of****the**pencil which meets its image**.****This**involves

**a**sifting

**of**

**the**empirical and

__rational__elements entering into

__each__social science statement

**.**

Kant argued

**for****the**establishment**of****a**peaceful world community**,**not**in****a**sense**of****a**global government**,****but****in****the**hope**that**__each__state would declare itself**a**free state**that**respects its citizens and welcomes foreign visitors as fellow__rational__beings**,**thus promoting peaceful society worldwide**.**
According

**to**Weber**,**Confucianism and Puritanism are mutually exclusive types**of**__rational__thought**,**__each__attempting**to**prescribe**a****way****of**life based on religious dogma**.**
As

**a**consequence**of****the**Weierstrass approximation theorem**,****one**can show**that****the**space C**is**separable:**the**polynomial functions are dense**,**and__each__polynomial function can**be**uniformly approximated by**one**with__rational__coefficients**;**there are only countably many polynomials with__rational__coefficients**.**
Since they are produced automatically without

**any**__rational__analysis and verification ( see**the**modern idea**of****the**subconscious )**of**whether they are correct or not**,**they need**to****be**confirmed ( epimarteresis: confirmation ),**a**process which must follow__each__assumption**.**
Therefore

**,****the**__rational__decision**for**__each__voter**is****to****be**generally ignorant**of**politics and perhaps even abstain from voting**.**
** An individual voter

**may**have**a**__rational__ignorance regarding politics**,**especially**in**nationwide elections**,**since__each__vote has little weight**.**
The

__rational__powers and abilities**of**__each__and every human being were attributed**to**his soul**,**which was**a**genius**.**
Field Marshal Viscount Alanbrooke

**,**Chief**of****the**Imperial General Staff and co-chairman**of****the**Anglo-US Combined Chiefs**of**Staff Committee**for**most**of****the**Second World War**,**described**the**art**of**military strategy as: "**to**derive from**the**aim**a**series**of**military objectives**to****be**achieved:**to**assess these objectives as**to****the**military requirements they create**,**and**the**pre-conditions which**the**achievement**of**__each__**is**likely**to**necessitate:**to**measure available and potential resources against**the**requirements and**to**chart from**this**process**a**coherent pattern**of**priorities and**a**__rational__course**of**action**.**
“

**to**derive from**the**aim**a**series**of**military objectives**to****be**achieved:**to**assess these objectives as**to****the**military requirements they create**,**and**the**pre-conditions which**the**achievement**of**__each__**is**likely**to**necessitate:**to**measure available and potential resources against**the**requirements and**to**chart from**this**process**a**coherent pattern**of**priorities and**a**__rational__course**of**action .”
EMH advocates reply

**that**while individual market participants do not always act rationally ( or have complete information ), their aggregate decisions balance__each__other**,**resulting**in****a**__rational__outcome ( optimists who buy stock and bid**the**price higher are countered by pessimists who sell their stock**,**which keeps**the**price**in**equilibrium ).
Hunter

**,**unlike his contemporaries … sought**the**reason**for**__each__phenomenon ),**but**because it afforded him**the**opportunity**,**given his empirical rather than__rational__bent**,****to**study his main interest-life**,****in****all**its forms**.**
Thus

**in****the**common context**of**polynomials with__rational__coefficients**,****a**polynomial**is**irreducible if it cannot**be**expressed as**the**product**of**two or more such polynomials**,**__each__**of**them having**a**lower degree than**the**original**one****.**
Scientists on

**the**ground will use X-ray crystallography**to**study__each__protein's three-dimensional structure which**,**when determined**,****may**aid**in**controlling__each__protein's activity**through**__rational__drug design**.**
Therefore

**the**problem**of**computing derivatives**,**antiderivatives**,**integrals**,**power series expansions**,**Fourier series**,**residues**,**and linear functional transformations**of**__rational__functions can**be**reduced**,**via partial fraction decomposition**,****to**making**the**computation on__each__single element used**in****the**decomposition**.**
However

**,**classical bargaining theory assumes**that**__each__participant**in****a**bargaining process will choose between possible agreements**,**following**the**conduct predicted by**the**__rational__choice model**.**
* Pseudo algebraically closed field ( mathematics ),

**a**field with geometric features**,**namely__each__variety over it has**a**__rational__point
The costs are allocated

**in****a**__rational__and systematic manner as depreciation expense**to**__each__period**in**which**the**asset**is**used**,**beginning when**the**asset**is**placed**in**service**.**
Lange has been called

**the**poetical theologian par excellence: “ It has been said**of**him**that**his thoughts succeed__each__other**in**such rapid and agitated waves**that****all**calm reflection and**all**__rational__distinction become**,****in****a**manner**,**drowned ” ( F**.**Lichtenberger ).
He defends

**the**Thomistic position**that**human beings are essentially__rational__animals**,**__each__**one**miraculously created**.**1.730 seconds.